Answer :
To solve this problem, we need to understand the concept of relativistic mass increase when an electron is accelerated to high speeds. The energy gained by the electron, when accelerated through a potential difference [tex]V[/tex], is given by the equation:
[tex]E = eV[/tex]
where:
- [tex]e[/tex] is the charge of the electron (approximately [tex]1.6 \times 10^{-19}[/tex] coulombs),
- [tex]V[/tex] is the potential difference (here, 10.2 million volts or [tex]10.2 \times 10^{6} \text{ V}[/tex]).
Thus, the gained energy is:
[tex]E = (1.6 \times 10^{-19} \text{ C}) \times (10.2 \times 10^{6} \text{ V}) = 1.632 \times 10^{-12} \text{ joules}[/tex]
The rest mass energy of an electron is given by the equation [tex]E_0 = mc^2[/tex], where [tex]m[/tex] denotes the rest mass of the electron ([tex]9.11 \times 10^{-31} \text{ kg}[/tex]) and [tex]c[/tex] is the speed of light [tex](3 \times 10^8 \text{ m/s})[/tex]. This calculates to:
[tex]E_0 = 9.11 \times 10^{-31} \text{ kg} \times (3 \times 10^8 \text{ m/s})^2 = 8.2 \times 10^{-14} \text{ joules}[/tex]
The relativistic mass or total energy [tex]E_t[/tex] after gaining energy is the sum of rest mass energy and the energy gained:
[tex]E_t = E_0 + E = 8.2 \times 10^{-14} \text{ J} + 1.632 \times 10^{-12} \text{ J}[/tex]
Simplifying gives:
[tex]E_t = 1.714 \times 10^{-12} \text{ joules}[/tex]
The percent increase in mass (or energy) is calculated by:
[tex]\text{Percent increase} = \left( \frac{E - E_0}{E_0} \right) \times 100
d[/tex]
Substituting the values, we have:
[tex]\text{Percent increase} = \left( \frac{1.632 \times 10^{-12} - 8.2 \times 10^{-14}}{8.2 \times 10^{-14}} \right) \times 100[/tex]
Solving this gives:
[tex]\text{Percent increase} \approx 2000[/tex]
Therefore, the correct answer is option [B] 2000.
